Egg-Shaped Orbits of Non-Uniform Oscillators

A simple non-uniform oscillator follows the differential equation (the blue curve). In an imaginary solar system where the planets follow this law of motion, but where angular momentum is conserved, planetary orbits would be egg-shaped (the red curve), with the parameter taking a role akin to the orbit eccentricity.

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We are able to plot both the angular velocity (blue curve) and distance from the origin (red curve) on the same axes for comparison by casting the equations in dimensionless form. For visual convenience we also define the functions so that when the oscillation is uniform, both the angular velocity and distance from the origin are equal to unity, as described below:

The curves are related by the need to keep the angular momentum equal to a constant, . As an equation we therefore have, . We define the angular velocity with (you can vary with the slider control), then define , and obtain as a result.

In the case where the non-uniformity is zero, we have constant angular velocity with an intuitively expected circular orbit. When the non-uniformity becomes significant at around 0.9, the angular velocity becomes very small in the region near . To compensate, the orbital radius grows very large, hence an egg-shaped orbit. As approaches unity, the orbital radius must approach infinity and the orbit is then no longer closed.